A355716 a(n) is the smallest number that has exactly n binary palindrome divisors (A006995).

1, 3, 9, 15, 99, 45, 135, 189, 315, 495, 945, 765, 2079, 6237, 3465, 5355, 4095, 8415, 31185, 20475, 25245, 12285, 85995, 58905, 61425, 45045, 69615, 176715, 446985, 225225, 328185, 208845, 135135, 405405, 528255, 1396395, 675675, 2027025, 765765, 5360355, 2993445, 3968055, 3828825

Offset: 1

Author

Bernard Schott, Jul 15 2022

Keywords

• nonn
• base

Examples

			a(4) = 15 since 15 has 4 divisors {1, 3, 5, 15} that are all palindromes when written in binary: 1, 11, 101 and 1111; no positive integer smaller than 15 has four divisors that are binary palindromes, hence a(4) = 15.
a(5) = 99 since 99 has 6 divisors {1, 3, 9, 11, 33, 99} of which only 11 is not a palindrome when written in binary: 11_10 = 1011_2; no positive integer smaller than 99 has five divisors that are binary palindromes, hence a(5) = 99.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..67

Crossrefs

Cf. A006995, A175242, A329419, A329420.

Programs

Mathematica

f[n_] := DivisorSum[n, 1 &, PalindromeQ[IntegerDigits[#, 2]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[25, 10^5] (* Amiram Eldar, Jul 15 2022 *)

PARI

is(n) = my(d=binary(n)); d==Vecrev(d); \\ A006995
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Jul 15 2022

Python

from sympy import divisors
from itertools import count, islice
def c(n): b = bin(n)[2:]; return b == b[::-1]
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 20))) # Michael S. Branicky, Jul 23 2022

Extensions

More terms from Michael S. Branicky, Jul 15 2022